Understanding how to apply Stokes theorem to higher dimensional spaces, non-Euclidean metrics, and with curvilinear coordinates has been a long standing goal.

A traditional answer to these questions can be found in the formalism of differential forms, as covered for example in [2], and [8]. However, both of those texts, despite their small size, are intensely scary. I also found it counter intuitive to have to express all physical quantities as forms, since there are many times when we don’t have any pressing desire to integrate these.

Later I encountered Denker’s straight wire treatment [1], which states that the geometric algebra formulation of Stokes theorem has the form

This is simple enough looking, but there are some important details left out. In particular the grades do not match, so there must be some sort of implied projection or dot product operations too. We also need to understand how to express the hypervolume and hypersurfaces when evaluating these integrals, especially when we want to use curvilinear coordinates.

I’d attempted to puzzle through these details previously. A collection of these attempts, to be removed from my collection of geometric algebra notes, can be found in [4]. I’d recently reviewed all of these and wrote a compact synopsis [5] of all those notes, but in the process of doing so, I realized there was a couple of fundamental problems with the approach I had used.

One detail that was that I failed to understand, was that we have a requirement for treating a infinitesimal region in the proof, then summing over such regions to express the boundary integral. Understanding that the boundary integral form and its dot product are both evaluated only at the end points of the integral region is an important detail that follows from such an argument (as used in proof of Stokes theorem for a 3D Cartesian space in [7].)

I also realized that my previous attempts could only work for the special cases where the dimension of the integration volume also equaled the dimension of the vector space. The key to resolving this issue is the concept of the tangent space, and an understanding of how to express the projection of the gradient onto the tangent space. These concepts are covered thoroughly in [6], which also introduces Stokes theorem as a special case of a more fundamental theorem for integration of geometric algebraic objects. My objective, for now, is still just to understand the generalization of Stokes theorem, and will leave the fundamental theorem of geometric calculus to later.

Now that these details are understood, the purpose of these notes is to detail the Geometric algebra form of Stokes theorem, covering its generalization to higher dimensional spaces and non-Euclidean metrics (i.e. especially those used for special relativity and electromagnetism), and understanding how to properly deal with curvilinear coordinates. This generalization has the form

Theorem 1. Stokes’ Theorem

For blades , and volume element ,

Here the volume integral is over a dimensional surface (manifold), is the projection of the gradient onto the tangent space of the manifold, and indicates integration over the boundary of .

It takes some work to give this more concrete meaning. I will attempt to do so in a gradual fashion, and provide a number of examples that illustrate some of the relevant details.

Basic notation

A finite vector space, not necessarily Euclidean, with basis will be assumed to be the generator of the geometric algebra. A dual or reciprocal basis for this basis can be calculated, defined by the property

This is an Euclidean space when .

To select from a multivector the grade portion, say we write

The scalar portion of a multivector will be written as

The grade selection operators can be used to define the outer and inner products. For blades , and of grade and respectively, these are

Written out explicitly for odd grade blades (vector, trivector, …), and vector the dot and wedge products are respectively

Similarly for even grade blades these are

It will be useful to employ the cyclic scalar reordering identity for the scalar selection operator

For an dimensional vector space, a product of orthonormal (up to a sign) unit vectors is referred to as a pseudoscalar for the space, typically denoted by

The pseudoscalar may commute or anticommute with other blades in the space. We may also form a pseudoscalar for a subspace spanned by vectors by unit scaling the wedge products of those vectors .

Curvilinear coordinates

For our purposes a manifold can be loosely defined as a parameterized surface. For example, a 2D manifold can be considered a surface in an dimensional vector space, parameterized by two variables

Note that the indices here do not represent exponentiation. We can construct a basis for the manifold as

On the manifold we can calculate a reciprocal basis , defined by requiring, at each point on the surface

Associated implicitly with this basis is a curvilinear coordinate representation defined by the projection operation

(sums over mixed indices are implied). These coordinates can be calculated by taking dot products with the reciprocal frame vectors

In this document all coordinates are with respect to a specific curvilinear basis, and not with respect to the standard basis or its dual basis unless otherwise noted.

Similar to the usual notation for derivatives with respect to the standard basis coordinates we form a lower index partial derivative operator

so that when the complete vector space is spanned by the gradient has the curvilinear representation

This can be motivated by noting that the directional derivative is defined by

When the basis does not span the space, the projection of the gradient onto the tangent space at the point of evaluation

This is called the vector derivative.

See [6] for a more complete discussion of the gradient and vector derivatives in curvilinear coordinates.

Green’s theorem

Given a two parameter () surface parameterization, the curvilinear coordinate representation of a vector has the form

We assume that the vector space is of dimension two or greater but otherwise unrestricted, and need not have an Euclidean basis. Here denotes the rejection of from the tangent space at the point of evaluation. Green’s theorem relates the integral around a closed curve to an “area” integral on that surface

Theorem 2. Green’s Theorem

Following the arguments used in [7] for Stokes theorem in three dimensions, we first evaluate the loop integral along the differential element of the surface at the point evaluated over the range , as shown in the infinitesimal loop of fig. 1.1.

Fig 1.1. Infinitesimal loop integral

Over the infinitesimal area, the loop integral decomposes into

where the differentials along the curve are

It is assumed that the parameterization change is small enough that this loop integral can be considered planar (regardless of the dimension of the vector space). Making use of the fact that for , the loop integral is

With the distances being infinitesimal, these differences can be rewritten as partial differentials

We can now sum over a larger area as in fig. 1.2

Fig 1.2. Sum of infinitesimal loops

All the opposing oriented loop elements cancel, so the integral around the complete boundary of the surface is given by the area integral of the partials difference.

We will see that Green’s theorem is a special case of the Curl (Stokes) theorem. This observation will also provide a geometric interpretation of the right hand side area integral of thm. 2, and allow for a coordinate free representation.

Special case:

An important special case of Green’s theorem is for a Euclidean two dimensional space where the vector function is

Here Green’s theorem takes the form

Curl theorem, two volume vector field

Having examined the right hand side of thm. 1 for the very simplest geometric object , let’s look at the right hand side, the area integral in more detail. We restrict our attention for now to vectors still defined by eq. 1.19.

First we need to assign a meaning to . By this, we mean the wedge products of the two differential elements. With

that area element is

This is the oriented area element that lies in the tangent plane at the point of evaluation, and has the magnitude of the area of that segment of the surface, as depicted in fig. 1.3.

Fig 1.3. Oriented area element tiling of a surface

Observe that we have no requirement to introduce a normal to the surface to describe the direction of the plane. The wedge product provides the information about the orientation of the place in the space, even when the vector space that our vector lies in has dimension greater than three.

Proceeding with the expansion of the dot product of the area element with the curl, using eq. 1.0.6, eq. 1.0.7, and eq. 1.0.8, and a scalar selection operation, we have

Let’s proceed to expand the inner dot product

The complete curl term is thus

This almost has the form of eq. 1.23, although that is not immediately obvious. Working backwards, using the shorthand , we can show that this coordinate representation can be eliminated

This relates the two parameter surface integral of the curl to the loop integral over its boundary

This is the very simplest special case of Stokes theorem. When written in the general form of Stokes thm. 1

we must remember (the is to remind us of this) that it is implied that both the vector and the differential elements are evaluated on the boundaries of the integration ranges respectively. A more exact statement is

Expanded out in full this is

which can be cross checked against fig. 1.4 to demonstrate that this specifies a clockwise orientation. For the surface with oriented area , the clockwise loop is designated with line elements (1)-(4), we see that the contributions around this loop (in boxes) match eq. 1.0.35.

For a Cartesian 2D Euclidean parameterization of a vector field and the integration space, Stokes theorem should be equivalent to Green’s theorem eq. 1.0.25. Let’s expand both sides of eq. 1.0.32 independently to verify equality. The parameterization is

Here the dual basis is the basis, and the projection onto the tangent space is just the gradient

The volume element is an area weighted pseudoscalar for the space

and the curl of a vector is

So, the LHS of Stokes theorem takes the coordinate form

For the RHS, following fig. 1.5, we have

As expected, we can also obtain this by integrating eq. 1.0.38.

Fig 1.5. Euclidean 2D loop

Example: Cylindrical parameterization

Let’s now consider a cylindrical parameterization of a 4D space with Euclidean metric or Minkowski metric . For such a space let’s do a brute force expansion of both sides of Stokes theorem to gain some confidence that all is well.

With , such a space is conveniently parameterized as illustrated in fig. 1.6 as

Fig 1.6. Cylindrical polar parameterization

Note that the Euclidean case where rejection of the non-axial components of expands to

whereas for the Minkowski case where we have a hyperbolic expansion

Within such a space consider the surface along , for which the vectors are parameterized by

The tangent space unit vectors are

and

Observe that both of these vectors have their origin at the point of evaluation, and aren’t relative to the absolute origin used to parameterize the complete space.

We wish to compute the volume element for the tangent plane. Noting that and both anticommute with we have for

so

The tangent space volume element is thus

With the tangent plane vectors both perpendicular we don’t need the general lemma 6 to compute the reciprocal basis, but can do so by inspection

and

Observe that the latter depends on the metric signature.

The vector derivative, the projection of the gradient on the tangent space, is

From this we see that acting with the vector derivative on a scalar radial only dependent function is a vector function that has a radial direction, whereas the action of the vector derivative on an azimuthal only dependent function is a vector function that has only an azimuthal direction. The interpretation of the geometric product action of the vector derivative on a vector function is not as simple since the product will be a multivector.

Expanding the curl in coordinates is messier, but yields in the end when tackled with sufficient care

After all this reduction, we can now state in coordinates the LHS of Stokes theorem explicitly

Now compare this to the direct evaluation of the loop integral portion of Stokes theorem. Expressing this using eq. 1.0.34, we have the same result

This example highlights some of the power of Stokes theorem, since the reduction of the volume element differential form was seen to be quite a chore (and easy to make mistakes doing.)

Example: Composition of boost and rotation

Working in a space with basis where and , an active composition of boost and rotation has the form

where is a bivector of a timelike unit vector and perpendicular spacelike unit vector, and is a bivector of two perpendicular spacelike unit vectors. For example, and . For such the respective Lorentz transformation matrices are

and

Let’s calculate the tangent space vectors for this parameterization, assuming that the particle is at an initial spacetime position of . That is

To calculate the tangent space vectors for this subspace we note that

and

The tangent space vectors are therefore

Continuing a specific example where let’s also pick , the spacetime position of a particle at the origin of a frame at that frame’s . The tangent space vectors for the subspace parameterized by this transformation and this initial position is then reduced to

and

By inspection the dual basis for this parameterization is

So, Stokes theorem, applied to a spacetime vector , for this subspace is

Since the point is to avoid the curl integral, we did not actually have to state it explicitly, nor was there any actual need to calculate the dual basis.

Example: Dual representation in three dimensions

It’s clear that there is a projective nature to the differential form . This projective nature allows us, in three dimensions, to re-express Stokes theorem using the gradient instead of the vector derivative, and to utilize the cross product and a normal direction to the plane.

When we parameterize a normal direction to the tangent space, so that for a 2D tangent space spanned by curvilinear coordinates and the vector is normal to both, we can write our vector as

and express the orientation of the tangent space area element in terms of a pseudoscalar that includes this normal direction

Inserting this into an expansion of the curl form we have

Observe that this last term, the contribution of the component of the gradient perpendicular to the tangent space, has no components

leaving

Now scale the normal vector and its dual to have unit norm as follows

so that for , the volume element can be

This scaling choice is illustrated in fig. 1.7, and represents the “outwards” normal. With such a scaling choice we have

Fig 1.7. Outwards normal

and almost have the desired cross product representation

With the duality identity , we have the traditional 3D representation of Stokes theorem

Note that the orientation of the loop integral in the traditional statement of the 3D Stokes theorem is counterclockwise instead of clockwise, as written here.

Stokes theorem, three variable volume element parameterization

We can restate the identity of thm. 1 in an equivalent dot product form.

Here , with the implicit assumption that it and the blade that it is dotted with, are both evaluated at the end points of integration variable that has been integrated against.

We’ve seen one specific example of this above in the expansions of eq. 1.28, and eq. 1.29, however, the equivalent result of eq. 1.0.78, somewhat magically, applies to any degree blade and volume element provided the degree of the blade is less than that of the volume element (i.e. ). That magic follows directly from lemma 1.

As an expositional example, consider a three variable volume element parameterization, and a vector blade

It should not be surprising that this has the structure found in the theory of differential forms. Using the differentials for each of the parameterization “directions”, we can write this dot product expansion as

Observe that the sign changes with each element of that is skipped. In differential forms, the wedge product composition of 1-forms is an abstract quantity. Here the differentials are just vectors, and their wedge product represents an oriented volume element. This interpretation is likely available in the theory of differential forms too, but is arguably less obvious.

Digression

As was the case with the loop integral, we expect that the coordinate representation has a representation that can be expressed as a number of antisymmetric terms. A bit of experimentation shows that such a sum, after dropping the parameter space volume element factor, is

To proceed with the integration, we must again consider an infinitesimal volume element, for which the partial can be evaluated as the difference of the endpoints, with all else held constant. For this three variable parameterization, say, , let’s delimit such an infinitesimal volume element by the parameterization ranges , , . The integral is

Extending this over the ranges , , , we have proved Stokes thm. 1 for vectors and a three parameter volume element, provided we have a surface element of the form

where the evaluation of the dot products with are also evaluated at the same points.

Example: Euclidean spherical polar parameterization of 3D subspace

Consider an Euclidean space where a 3D subspace is parameterized using spherical coordinates, as in

The tangent space basis for the subspace situated at some fixed , is easy to calculate, and is found to be

While we can use the general relation of lemma 7 to compute the reciprocal basis. That is

However, a naive attempt at applying this without algebraic software is a route that requires a lot of care, and is easy to make mistakes doing. In this case it is really not necessary since the tangent space basis only requires scaling to orthonormalize, satisfying for

This allows us to read off the dual basis for the tangent volume by inspection

Should we wish to explicitly calculate the curl on the tangent space, we would need these. The area and volume elements are also messy to calculate manually. This expansion can be found in the Mathematica notebook \nbref{sphericalSurfaceAndVolumeElements.nb}, and is

Those area elements have a Geometric algebra factorization that are perhaps useful

One of the beauties of Stokes theorem is that we don’t actually have to calculate the dual basis on the tangent space to proceed with the integration. For that calculation above, where we had a normal tangent basis, I still used software was used as an aid, so it is clear that this can generally get pretty messy.

To apply Stokes theorem to a vector field we can use eq. 1.0.82 to write down the integral directly

Observe that eq. 1.0.90 is a vector valued integral that expands to

This could easily be a difficult integral to evaluate since the vectors evaluated at the endpoints are still functions of two parameters. An easier integral would result from the application of Stokes theorem to a bivector valued field, say , for which we have

There is a geometric interpretation to these oriented area integrals, especially when written out explicitly in terms of the differentials along the parameterization directions. Pulling out a sign explicitly to match the geometry (as we had to also do for the line integrals in the two parameter volume element case), we can write this as

When written out in this differential form, each of the respective area elements is an oriented area along one of the faces of the parameterization volume, much like the line integral that results from a two parameter volume curl integral. This is visualized in fig. 1.8. In this figure, faces (1) and (3) are “top faces”, those with signs matching the tops of the evaluation ranges eq. 1.0.94, whereas face (2) is a bottom face with a sign that is correspondingly reversed.

Working with a three parameter volume element in a Minkowski space does not change much. For example in a 4D space with , we can employ a hyperbolic-spherical parameterization similar to that used above for the 4D Euclidean space

This has tangent space basis elements

This is a normal basis, but again not orthonormal. Specifically, for we have

where we see that the radial vector is timelike. We can form the dual basis again by inspection

The area elements are

or

The volume element also reduces nicely, and is

The area and volume element reductions were once again messy, done in software using \nbref{sphericalSurfaceAndVolumeElementsMinkowski.nb}. However, we really only need eq. 1.0.96 to perform the Stokes integration.

Stokes theorem, four variable volume element parameterization

Volume elements for up to four parameters are likely of physical interest, with the four volume elements of interest for relativistic physics in spaces. For example, we may wish to use a parameterization , with a four volume

We follow the same procedure to calculate the corresponding boundary surface “area” element (with dimensions of volume in this case). This is

Our boundary value surface element is therefore

where it is implied that this (and the dot products with ) are evaluated on the boundaries of the integration ranges of the omitted index. This same boundary form can be used for vector, bivector and trivector variations of Stokes theorem.

Duality and its relation to the pseudoscalar.

Looking to eq. 1.0.181 of lemma 6, and scaling the wedge product by its absolute magnitude, we can express duality using that scaled bivector as a pseudoscalar for the plane that spans . Let’s introduce a subscript notation for such scaled blades

This allows us to express the unit vector in the direction of as

Following the pattern of eq. 1.0.181, it is clear how to express the dual vectors for higher dimensional subspaces. For example

or for the unit vector in the direction of ,

Divergence theorem.

When the curl integral is a scalar result we are able to apply duality relationships to obtain the divergence theorem for the corresponding space. We will be able to show that a relationship of the following form holds

Here is a vector, is normal to the boundary surface, and is the area of this bounding surface element. We wish to quantify these more precisely, especially because the orientation of the normal vectors are metric dependent. Working a few specific examples will show the pattern nicely, but it is helpful to first consider some aspects of the general case.

First note that, for a scalar Stokes integral we are integrating the vector derivative curl of a blade over a k-parameter volume element. Because the dimension of the space matches the number of parameters, the projection of the gradient onto the tangent space is exactly that gradient

Multiplication of by the pseudoscalar will always produce a vector. With the introduction of such a dual vector, as in

Stokes theorem takes the form

or

where we will see that the vector can roughly be characterized as a normal to the boundary surface. Using primes to indicate the scope of the action of the gradient, cyclic permutation within the scalar selection operator can be used to factor out the pseudoscalar

The second last step uses lemma 8, and the last writes , where we have assumed (without loss of generality) that has the same orientation as the pseudoscalar for the space. We also assume that the parameterization is non-degenerate over the integration volume (i.e. no ), so the sign of this product cannot change.

Let’s now return to the normal vector . With (the indexed differential omitted), and , we have

We’ve seen in eq. 1.0.106 and lemma 7 that the dual of vector with respect to the unit pseudoscalar in a subspace spanned by is

or

This allows us to write

where , and is the area of the boundary area element normal to . Note that the term will now cancel cleanly from both sides of the divergence equation, taking both the metric and the orientation specific dependencies with it.

This leaves us with

To spell out the details, we have to be very careful with the signs. However, that is a job best left for specific examples.

Example: 2D divergence theorem

Let’s start back at

On the left our integral can be rewritten as

where and we pick the pseudoscalar with the same orientation as the volume (area in this case) element .

For the boundary form we have

The duality relations for the tangent space are

or

Back substitution into the line element gives

Writing (no sum) , we have

This provides us a divergence and normal relationship, with terms on each side that can be canceled. Restoring explicit range evaluation, that is

Let’s consider this graphically for an Euclidean metric as illustrated in fig. 1.9.

Fig 1.9. Normals on area element

We see that

along the outwards normal is ,

along the outwards normal is ,

along the outwards normal is , and

along the outwards normal is .

Writing that outwards normal as , we have

Note that we can use the same algebraic notion of outward normal for non-Euclidean spaces, although cannot expect the geometry to look anything like that of the figure.

Example: 3D divergence theorem

As with the 2D example, let’s start back with

In a 3D space, the pseudoscalar commutes with all grades, so we have

where , and we have used a pseudoscalar with the same orientation as the volume element

In the boundary integral our dual two form is

where , and

Observe that we can do a cyclic permutation of a 3 blade without any change of sign, for example

Because of this we can write the dual two form as we expressed the normals in lemma 7

We can now state the 3D divergence theorem, canceling out the metric and orientation dependent term on both sides

where (sums implied)

and

The outwards normals at the upper integration ranges of a three parameter surface are depicted in fig. 1.10.

Fig 1.10. Outwards normals on volume at upper integration ranges.

This sign alternation originates with the two form elements from the Stokes boundary integral, which were explicitly evaluated at the endpoints of the integral. That is, for ,

In the context of the divergence theorem, this means that we are implicitly requiring the dot products to be evaluated specifically at the end points of the integration where , accounting for the alternation of sign required to describe the normals as uniformly outwards.

Example: 4D divergence theorem

Applying Stokes theorem to a trivector in the 4D case we find

Here the pseudoscalar has been picked to have the same orientation as the hypervolume element . Writing the dual of the three form is

Here, we’ve written

Observe that the dual representation nicely removes the alternation of sign that we had in the Stokes theorem boundary integral, since each alternation of the wedged vectors in the pseudoscalar changes the sign once.

As before, we define the outwards normals as on the upper and lower integration ranges respectively. The scalar area elements on these faces can be written in a dual form

so that the 4D divergence theorem looks just like the 2D and 3D cases

Here we define the volume scaled normal as

As before, we have made use of the implicit fact that the three form (and it’s dot product with ) was evaluated on the boundaries of the integration region, with a toggling of sign on the lower limit of that evaluation that is now reflected in what we have defined as the outwards normal.

We also obtain explicit instructions from this formalism how to compute the “outwards” normal for this surface in a 4D space (unit scaling of the dual basis elements), something that we cannot compute using any sort of geometrical intuition. For free we’ve obtained a result that applies to both Euclidean and Minkowski (or other non-Euclidean) spaces.

Volume integral coordinate representations

It may be useful to formulate the curl integrals in tensor form. For vectors , and bivectors , the coordinate representations of those differential forms (\cref{pr:stokesTheoremGeometricAlgebraII:1}) are

Here the bivector and trivector is expressed in terms of their curvilinear components on the tangent space

where

For the trivector components are also antisymmetric, changing sign with any interchange of indices.

Note that eq. 1.0.144d and eq. 1.0.144f appear much different on the surface, but both have the same structure. This can be seen by writing for former as

In both of these we have an alternation of sign, where the tensor index skips one of the volume element indices is sequence. We’ve seen in the 4D divergence theorem that this alternation of sign can be related to a duality transformation.

In integral form (no sum over indexes in terms), these are

Of these, I suspect that only eq. 1.0.148a and eq. 1.0.148d are of use.

Final remarks

Because we have used curvilinear coordinates from the get go, we have arrived naturally at a formulation that works for both Euclidean and non-Euclidean geometries, and have demonstrated that Stokes (and the divergence theorem) holds regardless of the geometry or the parameterization. We also know explicitly how to formulate both theorems for any parameterization that we choose, something much more valuable than knowledge that this is possible.

For the divergence theorem we have introduced the concept of outwards normal (for example in 3D, eq. 1.0.136), which still holds for non-Euclidean geometries. We may not be able to form intuitive geometrical interpretations for these normals, but do have an algebraic description of them.

Appendix

Problems

Question: Expand volume elements in coordinates

Show that the coordinate representation for the volume element dotted with the curl can be represented as a sum of antisymmetric terms. That is

(a)Prove eq. 1.0.144a

(b)Prove eq. 1.0.144b

(c)Prove eq. 1.0.144c

(d)Prove eq. 1.0.144d

(e)Prove eq. 1.0.144e

(f)Prove eq. 1.0.144f

Answer

(a) Two parameter volume, curl of vector

(b) Three parameter volume, curl of vector

(c) Four parameter volume, curl of vector

(d) Three parameter volume, curl of bivector

(e) Four parameter volume, curl of bivector

To start, we require lemma 3. For convenience lets also write our wedge products as a single indexed quantity, as in for . The expansion is

This last step uses an intermediate result from the eq. 1.0.152 expansion above, since each of the four terms has the same structure we have previously observed.

(f) Four parameter volume, curl of trivector

Using the shorthand again, the initial expansion gives

Applying lemma 4 to expand the inner products within the braces we have

We can cancel those last terms using lemma 5. Using the same reverse chain rule expansion once more we have

or

The final result follows after permuting the indices slightly.

Some helpful identities

Lemma 1. Distribution of inner products

Given two blades with grades subject to , and a vector , the inner product distributes according to

This will allow us, for example, to expand a general inner product of the form .

The proof is straightforward, but also mechanical. Start by expanding the wedge and dot products within a grade selection operator

Solving for in

we have

The last term above is zero since we are selecting the grade element of a multivector with grades and , which has no terms for . Now we can expand the multivector product, for

The latter multivector (with the wedge product factor) above has grades and , so this selection operator finds nothing. This leaves

The first dot products term has grade and is selected, whereas the wedge term has grade (for ).

Lemma 2. Distribution of two bivectors

For vectors , , and bivector , we have

Proof follows by applying the scalar selection operator, expanding the wedge product within it, and eliminating any of the terms that cannot contribute grade zero values

Lemma 3. Inner product of trivector with bivector

Given a bivector , and trivector where and are vectors, the inner product is

This is also problem 1.1(c) from Exercises 2.1 in [3], and submits to a dumb expansion in successive dot products with a final regrouping. With

Lemma 4. Distribution of two trivectors

Given a trivector and three vectors , and , the entire inner product can be expanded in terms of any successive set inner products, subject to change of sign with interchange of any two adjacent vectors within the dot product sequence

To show this, we first expand within a scalar selection operator

Now consider any single term from the scalar selection, such as the first. This can be reordered using the vector dot product identity

The vector-trivector product in the latter grade selection operation above contributes only bivector and quadvector terms, thus contributing nothing. This can be repeated, showing that

Substituting this back into eq. 1.0.168 proves lemma 4.

Lemma 5. Permutation of two successive dot products with trivector

Given a trivector and two vectors and , alternating the order of the dot products changes the sign

This and lemma 4 are clearly examples of a more general identity, but I’ll not try to prove that here. To show this one, we have

Cancellation of terms above was because they could not contribute to a grade one selection. We also employed the relation for bivector and vector .

Lemma 6. Duality in a plane

For a vector , and a plane containing and , the dual of this vector with respect to this plane is

Satisfying

and

To demonstrate, we start with the expansion of

Dotting with we have

but dotting with yields zero

To complete the proof, we note that the product in eq. 1.177 is just the wedge squared

This duality relation can be recast with a linear denominator

or

We can use this form after scaling it appropriately to express duality in terms of the pseudoscalar.

Lemma 7. Dual vector in a three vector subspace

In the subspace spanned by , the dual of is

Consider the dot product of with .

The canceled term is eliminated since it is the product of a vector and trivector producing no scalar term. Substituting , and noting that , we have

Lemma 8. Pseudoscalar selection

For grade blade (i.e. a pseudoscalar), and vectors , the grade selection of this blade sandwiched between the vectors is

To show this, we have to consider even and odd grades separately. First for even we have

Obsolete with potential errors.

This post may be in error. I wrote this before understanding that the gradient used in Stokes Theorem must be projected onto the tangent space of the parameterized surface, as detailed in Alan MacDonald’s Vector and Geometric Calculus.

Original Post:

Motivation.

With the divergence of the energy momentum tensor converted from a volume to a surface integral given by

I got to wondering what a closed form algebraic expression for this curious (and foreign seeming) quantity was. It obviously must be related to the normal to the surface. It seemed to me that a natural way to answer this question was to consider this divergence integral over an arbitrarily parametrized volume. This turns out to be overkill, but a useful seeming digression.

A generally parametrized parallelepiped volume element.

Suppose we parametrize a volume by specifying that all the points in that volume are covered by the position vector from the origin, given by

At any point in the volume of interest, we can create a level curve, holding two of the parameters constant, and varying the remaining one. In particular, we can construct three direction vectors along these level curves, one for each parameter not held constant

The span of these vectors, provided they are non-degenerate, forms a parallelepiped, the volume of which is

This volume element can be expanded in a number of ways

where the Jacobian determinant is given by

Provided we are interested in a volume for which the sign of this Jacobian determinant does not change sign, our task is to evaluate and reduce the integral

to a set (and sum of) two dimensional integrals.

On the geometry of the surfaces.

Suppose that we integrate over the ranges , , . Observe that the outwards normals along the face is . This is

Similarly our normal on the face is

and on the face the outward normal is

Along the faces these are just negated. We can summarize these as

Expansion of the Jacobian determinant

Suppose, to start with, our divergence volume integral 1.8 has just the following term

The specifics of how the scalar is indexed will not matter yet, so let’s suppress it. The Jacobian determinant can be expanded along the column for

Performing the same task (really just performing cyclic permutation of indexes) we can now construct the whole divergence integral

Regrouping we have

Observe that we can factor these sums utilizing the normals for the parallelepiped volume element

Let’s look at the first of these integrals in more detail. We integrate the values of the evaluated on the points of the surface for which . To perform this integral we dot against the outward normal area element . We do the same, but subtract the integral where is evaluated on the surface , where we dot with the area element that has the inwards normal direction on that surface. This is then done for each of the surfaces of the parallelepiped that we are integrating over.

In terms of the outwards (area scaled) normals on the and surfaces respectively we can write

In each case, for the surfaces, our negated inwards normal form can be redefined so that we integrate over only the outwards normal directions, and we can use the oriented integral notation

To encode (or imply) whether we require a positive or negative sign on the area element tensor of 1.15 for the surface in question.

A look back, and looking forward.

Now, having performed this long winded calculation, the meaning of becomes clear. What’s also clear is how this could have been arrived at directly utilizing the divergence theorem in its normal vector form. We had only to re-write our equation as a vector equation in terms of the gradient

From this we see directly that .

Despite there being an easier way to find the form of , I still consider this a worthwhile exercise. It hints how one could generalize the arguments to the higher dimensional cases. The main task would be to construct the normals to the hypersurfaces bounding the hypervolume, and how to do this algebraically utilizing determinants may not be too hard (since we want a Jacobian determinant as the hypervolume element in the “volume” integral). We also got more than the normal physics text book proof of the divergence theorem for Cartesian coordinates, and did it here for a general parametrization. This wasn’t a complete argument since we didn’t consider a general surface, broken down into a triangular mesh. We really want volume elements with triangular sides instead of parallelograms.

Disclaimer.

This problem set is as yet ungraded (although only the second question will be graded).

Problem 1. Fun with , , , and the duality of Maxwell’s equations in vacuum.

1. Statement. rank 3 spatial antisymmetric tensor identities.

Prove that

and use it to find the familiar relation for

Also show that

(Einstein summation implied all throughout this problem).

1. Solution

We can explicitly expand the (implied) sum over indexes . This is

For any only one term is non-zero. For example with , we have just a contribution from the part of the sum

The value of this for is

whereas for we have

Our sum has value one when matches , and value minus one for when are permuted. We can summarize this, by saying that when we have

However, observe that when the RHS is

as desired, so this form works in general without any qualifier, completing this part of the problem.

This gives us

We have one more identity to deal with.

We can expand out this (implied) sum slow and dumb as well

Now, observe that for any only one term of this sum is picked up. For example, with no loss of generality, pick . We are left with only

This has the value

when and is zero otherwise. We can therefore summarize the evaluation of this sum as

completing this problem.

2. Statement. Determinant of three by three matrix.

Prove that for any matrix : and that .

2. Solution

In class Simon showed us how the first identity can be arrived at using the triple product . It occurred to me later that I’d seen the identity to be proven in the context of Geometric Algebra, but hadn’t recognized it in this tensor form. Basically, a wedge product can be expanded in sums of determinants, and when the dimension of the space is the same as the vector, we have a pseudoscalar times the determinant of the components.

For example, in , let’s take the wedge product of a pair of vectors. As preparation for the relativistic case We won’t require an orthonormal basis, but express the vector in terms of a reciprocal frame and the associated components

where

When we get to the relativistic case, we can pick (but don’t have to) the standard basis

for which our reciprocal frame is implicitly defined by the metric

Anyways. Back to the problem. Let’s examine the case. Our wedge product in coordinates is

Since there are only two basis vectors we have

Our wedge product is a product of the determinant of the vector coordinates, times the pseudoscalar .

This doesn’t look quite like the relation that we want to prove, which had an antisymmetric tensor factor for the determinant. Observe that we get the determinant by picking off the component of the bivector result (the only component in this case), and we can do that by dotting with . To get an antisymmetric tensor times the determinant, we have only to dot with a different pseudoscalar (one that differs by a possible sign due to permutation of the indexes). That is

Now, if we write and we have

We can write this in two different ways. One of which is

and the other of which is by introducing free indexes for and , and summing antisymmetrically over these. That is

So, we have

This result hold regardless of the metric for the space, and does not require that we were using an orthonormal basis. When the metric is Euclidean and we have an orthonormal basis, then all the indexes can be dropped.

The and cases follow in exactly the same way, we just need more vectors in the wedge products.

For the case we have

Again, with and , and we have

and we can choose to write this in either form, resulting in the identity

The case follows exactly the same way, and we have

This time with and , and , and we have

This one is almost the identity to be established later in problem 1.4. We have only to raise and lower some indexes to get that one. Note that in the Minkowski standard basis above, because must be a permutation of for a non-zero result, we must have

So raising and lowering the identity above gives us

No sign changes were required for the indexes , since they are paired.

Until we did the raising and lowering operations here, there was no specific metric required, so our first result 2.33 is the more general one.

There’s one more part to this problem, doing the antisymmetric sums over the indexes . For the case we have

We conclude that

For the case we have the same operation

So we conclude

It’s clear what the pattern is, and if we evaluate the sum of the antisymmetric tensor squares in we have

So, for our SR case we have

This was part of question 1.4, albeit in lower index form. Here since all indexes are matched, we have the same result without major change

The main difference is that we are now taking the determinant of a lower index tensor.

3. Statement. Rotational invariance of 3D antisymmetric tensor

Use the previous results to show that is invariant under rotations.

3. Solution

We apply transformations to coordinates (and thus indexes) of the form

With our tensor transforming as its indexes, we have

We’ve got 2.32, which after dropping indexes, because we are in a Euclidean space, we have

Let , which gives us

but since , we have shown that is invariant under rotation.

4. Statement. Rotational invariance of 4D antisymmetric tensor

Use the previous results to show that is invariant under Lorentz transformations.

4. Solution

This follows the same way. We assume a transformation of coordinates of the following form

5. Solution

Show that is invariant under Lorentz transformations. Consider the polynomial of , also called the characteristic polynomial of the matrix . Find the coefficients of the expansion of in powers of in terms of the components of . Use the result to argue that and are Lorentz invariant.

6. Solution

The invariance of the determinant

Let’s consider how any lower index rank 2 tensor transforms. Given a transformation of coordinates

where , and . Let’s reflect briefly on why this determinant is unit valued. We have

which implies that the transformation product is

the identity matrix. The identity matrix has unit determinant, so we must have

Since we have

which is all that we can say about the determinant of this class of transformations by considering just invariance. If we restrict the transformations of coordinates to those of the same determinant sign as the identity matrix, we rule out reflections in time or space. This seems to be the essence of the labeling.

Why dwell on this? Well, I wanted to be clear on the conventions I’d chosen, since parts of the course notes used , and , and gave that matrix unit determinant. That looks like it is equivalent to my , except that the one in the course notes is loose when it comes to lower and upper indexes since it gives .

I’ll write

and require this (not ) to be the matrix with unit determinant. Having cleared the index upper and lower confusion I had trying to reconcile the class notes with the rules for index manipulation, let’s now consider the Lorentz transformation of a lower index rank 2 tensor (not necessarily antisymmetric or symmetric)

We have, transforming in the same fashion as a lower index coordinate four vector (but twice, once for each index)

The determinant of the transformation tensor is

We see that the determinant of a lower index rank 2 tensor is invariant under Lorentz transformation. This would include our characteristic polynomial .

Expanding the determinant.

Utilizing 2.39 we can now calculate the characteristic polynomial. This is

However, , or . This means we have

and our determinant is reduced to

If we expand this out we have our powers of coefficients are

By 2.39 the coefficient is just .

The terms can be seen to be zero. For example, the first one is

where the final equality to zero comes from summing a symmetric and antisymmetric product.

Similarly the coefficients can be shown to be zero. Again the first as a sample is

Disregarding the factor, let’s just expand this antisymmetric sum

Of the two terms above that were retained, they are the only ones without a zero factor. Consider the first part of this remaining part of the sum. Employing the metric tensor, to raise indexes so that the antisymmetry of can be utilized, and then finally relabeling all the dummy indexes we have

This is just the negative of the second term in the sum, leaving us with zero.

Finally, we have for the coefficient ()

Therefore, our characteristic polynomial is

Observe that in matrix form our strength tensors are

From these we can compute easily by inspection

Computing the determinant is not so easy. The dumb and simple way of expanding by cofactors takes two pages, and yields eventually

That supplies us with a relation for the characteristic polynomial in and

Observe that we found this for the special case where and were perpendicular in homework 2. Observe that when we have that perpendicularity, we can solve for the eigenvalues by inspection

and were able to diagonalize the matrix to solve the Lorentz force equation in parametric form. When we had real eigenvalues and an orthogonal diagonalization when . For the , we had a two purely imaginary eigenvalues, and when this was a Hermitian diagonalization. For the general case, when one of , or was zero, things didn’t have the same nice closed form solution.

In general our eigenvalues are

For the purposes of this problem we really only wish to show that and are Lorentz invariants. When we have , a Lorentz invariant. This must mean that is itself a Lorentz invariant. Since that is invariant, and we require to be invariant for any other possible values of , the difference must also be Lorentz invariant.

7. Statement. Show that the pseudoscalar invariant has only boundary effects.

Use integration by parts to show that only depends on the values of at the “boundary” of spacetime (e.g. the “surface” depicted on page 105 of the notes) and hence does not affect the equations of motion for the electromagnetic field.

7. Solution

This proceeds in a fairly straightforward fashion

Now, observe that by the Bianchi identity, this second term is zero

Now we have a set of perfect differentials, and can integrate

We are left with a only contributions to the integral from the boundary terms on the spacetime hypervolume, three-volume normals bounding the four-volume integration in the original integral.

8. Statement. Electromagnetic duality transformations.

Show that the Maxwell equations in vacuum are invariant under the transformation: , where is the dual electromagnetic stress tensor. Replacing with is known as “electric-magnetic duality”. Explain this name by considering the transformation in terms of and . Are the Maxwell equations with sources invariant under electric-magnetic duality transformations?

8. Solution

Let’s first consider the explanation of the name. First recall what the expansions are of and in terms of and . These are

with , and .

The magnetic field components are

with and .

Now let’s expand the dual tensors. These are

and

Summarizing we have

Is there a sign error in the result? Other than that we have the same sort of structure for the tensor with and switched around.

Let’s write these in matrix form, to compare

From these we can see by inspection that we have

This is consistent with the stated result in [1] (except for a factor of due to units differences), so it appears the signs above are all kosher.

Now, let’s see if the if the dual tensor satisfies the vacuum equations.

So the first checks out, provided we have no sources. If we have sources, then we see here that Maxwell’s equations do not hold since this would imply that the four current density must be zero.

How about the Bianchi identity? That gives us

The factor of two is slightly curious. Is there a mistake above? If there is a mistake, it doesn’t change the fact that Maxwell’s equation

Gives us zero for the Bianchi identity under source free conditions of .

Problem 2. Transformation properties of and , again.

1. Statement

Use the form of from page 82 in the class notes, the transformation law for given further down that same page, and the explicit form of the matrix (say, corresponding to motion in the positive direction with speed ) to derive the transformation law of the fields and . Use the transformation law to find the electromagnetic field of a charged particle moving with constant speed in the positive direction and check that the result agrees with the one that you obtained in Homework 2.

1. Solution

Given a transformation of coordinates

our rank 2 tensor transforms as

Introducing matrices

and noting that , we can express the electromagnetic strength tensor transformation as

The class notes use , which violates our conventions on mixed upper and lower indexes, but the end result 3.80 is the same.

Writing

we can compute the transformed field strength tensor

As a check we have the antisymmetry that is expected. There is also a regularity to the end result that is aesthetically pleasing, hinting that things are hopefully error free. In coordinates for and this is

Writing , we have

which puts us enroute to a tidier vector form

For a vector , write , , allowing a compact description of the field transformation

Now, we want to consider the field of a moving particle. In the particle’s (unprimed) rest frame the field due to its potential is

Coordinates for a “stationary” observer, who sees this particle moving along the x-axis at speed are related by a boost in the direction

Therefore the fields in the observer frame will be

More explicitly with this is

Comparing to Problem 3 in Problem set 2, I see that this matches the result obtained by separately transforming the gradient, the time partial, and the scalar potential. Actually, if I am being honest, I see that I made a sign error in all the coordinates of when I initially did (this ungraded problem) in problem set 2. That sign error should have been obvious by considering the case which would have mysteriously resulted in inversion of all the coordinates of the observed electric field.

2. Statement

A particle is moving with velocity in perpendicular and fields, all given in some particular “stationary” frame of reference.

\begin{enumerate}
\item Show that there exists a frame where the problem of finding the particle trajectory can be reduced to having either only an electric or only a magnetic field.
\item Explain what determines which case takes place.
\item Find the velocity of that frame relative to the “stationary” frame.
\end{enumerate}

2. Solution

\paragraph{Part 1 and 2:} Existence of the transformation.

In the single particle Lorentz trajectory problem we wish to solve

which in matrix form we can write as

where we write our column vector proper velocity as . Under transformation of coordinates , with , this becomes

Suppose we can find eigenvectors for the matrix . That is for some eigenvalue , we can find an eigenvector

Rearranging we have

and conclude that lies in the null space of the matrix and that this difference of matrices must have a zero determinant

Since for any Lorentz transformation in , and we have

In problem 1.6, we called this our characteristic equation . Observe that the characteristic equation is Lorentz invariant for any , which requires that the eigenvalues are also Lorentz invariants.

In problem 1.6 of this problem set we computed that this characteristic equation expands to

The eigenvalues for the system, also each necessarily Lorentz invariants, are

Observe that in the specific case where , as in this problem, we must have in all frames, and the two non-zero eigenvalues of our characteristic polynomial are simply

These and are the invariants for this system. If we have in one frame, we must also have in another frame, still maintaining perpendicular fields. In particular if we maintain real eigenvalues. Similarly if in some frame, we must always have imaginary eigenvalues, and this is also true in the case.

While the problem can be posed as a pure diagonalization problem (and even solved numerically this way for the general constant fields case), we can also work symbolically, thinking of the trajectories problem as simply seeking a transformation of frames that reduce the scope of the problem to one that is more tractable. That does not have to be the linear transformation that diagonalizes the system. Instead we are free to transform to a frame where one of the two fields or is zero, provided the invariants discussed are maintained.

\paragraph{Part 3:} Finding the boost velocity that wipes out one of the fields.

Let’s now consider a Lorentz boost , and seek to solve for the boost velocity that wipes out one of the fields, given the invariants that must be maintained for the system

To make things concrete, suppose that our perpendicular fields are given by and .

Let also assume that we can find the velocity for which one or more of the transformed fields is zero. Suppose that velocity is

where are the direction cosines of so that . We will want to compute the components of and parallel and perpendicular to this velocity.

Those are

For the magnetic field we have

and

Now, observe that , but this is just zero. So we have . So our cross products terms are just

We can now express how the fields transform, given this arbitrary boost velocity. From 3.97, this is

Zero Electric field case.

Let’s tackle the two cases separately. First when , we can transform to a frame where . In coordinates from 3.117 this supplies us three sets of equations. These are

With an assumed solution the coordinate equation implies that one of or is zero. Perhaps there are solutions with too, but inspection shows that nicely kills off the first equation. Since , that also implies that we are left with

Or

Our velocity was solving the problem for the case up to an adjustable constant . That constant comes with constraints however, since we must also have our cosine . Expressed another way, the magnitude of the boost velocity is constrained by the relation

It appears we may also pick the equality case, so one velocity (not unique) that should transform away the electric field is

This particular boost direction is perpendicular to both fields. Observe that this highlights the invariance condition since we see this is required for a physically realizable velocity. Boosting in this direction will reduce our problem to one that has only the magnetic field component.

Zero Magnetic field case.

Now, let’s consider the case where we transform the magnetic field away, the case when our characteristic polynomial has strictly real eigenvalues . In this case, if we write out our equations for the transformed magnetic field and require these to separately equal zero, we have

Similar to before we see that kills off the first and second equations, leaving just

We now have a solution for the family of direction vectors that kill the magnetic field off

In addition to the initial constraint that , we have as before, constraints on the allowable values of

Like before we can pick the equality , yielding a boost direction of

Again, we see that the invariance condition is required for a physically realizable velocity if that velocity is entirely perpendicular to the fields.

Statement

Show explicitly that the electromagnetic 4-current for a particle moving with constant velocity (considered in class, p. 100-101 of notes) is conserved . Give a physical interpretation of this conservation law, for example by integrating over some spacetime region and giving an integral form to the conservation law ( is known as the “continuity equation”).

Solution

First lets review. Our four current was defined as

If each of the trajectories represents constant motion we have

The spacetime split of this four vector is

with differentials

Writing out the delta functions explicitly we have

So our time and space components of the current can be written

Each of these integrals can be evaluated with respect to the time coordinate delta function leaving the distribution

With this more general expression (multi-particle case) it should be possible to show that the four divergence is zero, however, the problem only asks for one particle. For the one particle case, we can make things really easy by taking the initial point in space and time as the origin, and aligning our velocity with one of the coordinates (say ).

Doing so we have the result derived in class

Our divergence then has only two portions

and these cancel out when summed. Note that this requires us to be loose with our delta functions, treating them like regular functions that are differentiable.

For the more general multiparticle case, we can treat the sum one particle at a time, and in each case, rotate coordinates so that the four divergence only picks up one term.

As for physical interpretation via integral, we have using the four dimensional divergence theorem

where is the three-volume element perpendicular to a plane. These volume elements are detailed generally in the text [2], however, they do note that one special case specifically , the element of the three-dimensional (spatial) volume “normal” to hyperplanes .

Without actually computing the determinants, we have something that is roughly of the form

This is cheating a bit to just write . Are there specific orientations required by the metric. To be precise we’d have to calculate the determinants detailed in the text, and then do the duality transformations.

Per unit time, we can write instead

Rather loosely this appears to roughly describe that the rate of change of charge in a volume must be matched with the “flow” of current through the surface within that amount of time.

A complex (phasor) representation is implied, so taking real parts when all is said and done is required of the fields. For the energy momentum tensor the Geometric Algebra form, modified for complex fields, is used

The assumed four vector potential will be written

Subject to the requirement that is a solution of Maxwell’s equation

To avoid latex hell, no special notation will be used for the Fourier coefficients,

When convenient and unambiguous, this dependence will be implied.

Having picked a time and space representation for the field, it will be natural to express both the four potential and the gradient as scalar plus spatial vector, instead of using the Dirac basis. For the gradient this is

and for the four potential (or the Fourier transform functions), this is

Setup

The field bivector is required for the energy momentum tensor. This is

This last term is a spatial curl and the field is then

Applied to the Fourier representation this is

The energy momentum tensor is then

The tensor integrated over all space. Energy and momentum?

Integrating this over all space and identification of the delta function

reduces the tensor to a single integral in the continuous angular wave number space of .

Observing that commutes with spatial bivectors and anticommutes with spatial vectors, and writing , one has

The scalar and spatial vector grade selection operator has been added for convenience and does not change the result since those are necessarily the only grades anyhow. The post multiplication by the observer frame time basis vector serves to separate the energy and momentum like components of the tensor nicely into scalar and vector aspects. In particular for , one could write

If these are correctly identified with energy and momentum then it also ought to be true that we have the conservation relationship

However, multiplying out (3.12) yields for

The vector component takes a bit more work to reduce

Canceling and regrouping leaves

This has no explicit dependence, so the conservation relation (3.14) is violated unless . There is no reason to assume that will be the case. In the discrete Fourier series treatment, a gauge transformation allowed for elimination of , and this implied or constant. We will probably have a similar result here, eliminating most of the terms in (3.15) and (3.16). Except for the constant solution of the field equations there is no obvious way that such a simplified energy expression will have zero derivative.

A more reasonable conclusion is that this approach is flawed. We ought to be looking at the divergence relation as a starting point, and instead of integrating over all space, instead employing Gauss’s theorem to convert the divergence integral into a surface integral. Without math, the conservation relationship probably ought to be expressed as energy change in a volume is matched by the momentum change through the surface. However, without an integral over all space, we do not get the nice delta function cancellation observed above. How to proceed is not immediately clear. Stepping back to review applications of Gauss’s theorem is probably a good first step.

This is a doubly complex representation, with the four vector pseudoscalar acting as a non-commutatitive imaginary, as well as real and imaginary parts for the electric and magnetic field vectors. We take the real part (not the scalar part) of any bivector solution of Maxwell’s equation as the actual solution, but allow ourself the freedom to work with the complex phasor representation when convenient. In these phasor vectors, the imaginary , as in , is a commuting imaginary, commuting with all the multivector elements in the algebra.

The real valued, four vector, energy momentum tensor was found to be

To supply some context that gives meaning to this tensor the associated conservation relationship was found to be

and in particular for , this four vector divergence takes the form

relating the energy term and the Poynting spatial vector with the current density and electric field product that constitutes the energy portion of the Lorentz force density.

Let’s apply this to calculating the energy associated with the field that is periodic within a rectangular prism as done by Bohm in [2]. We do not necessarily need the Geometric Algebra formalism for this calculation, but this will be a fun way to attempt it.

Setup

Let’s assume a Fourier representation for the four vector potential for the field . That is

where summation is over all angular wave number triplets . The Fourier coefficients are allowed to be complex valued, as is the resulting four vector , and the associated bivector field .

Fourier inversion, with , follows from

but only this orthogonality relationship and not the Fourier coefficients themselves

will be of interest here. Evaluating the curl for this potential yields

Since the four vector potential has been expressed using an explicit split into time and space components it will be natural to re express the bivector field in terms of scalar and (spatial) vector potentials, with the Fourier coefficients. Writing for the spatial basis vectors, , and , this is

The Faraday bivector field is then

This is now enough to express the energy momentum tensor

It will be more convenient to work with a scalar plus bivector (spatial vector) form of this tensor, and right multiplication by produces such a split

The primary object of this treatment will be consideration of the components of the tensor, which provide a split into energy density , and Poynting vector (momentum density) .

Our first step is to integrate (12) over the volume . This integration and the orthogonality relationship (6), removes the exponentials, leaving

Because commutes with the spatial bivectors, and anticommutes with the spatial vectors, the remainder of the Dirac basis vectors in these expressions can be eliminated

Expanding the energy momentum tensor components.

Energy

In (15) only the bivector-bivector and vector-vector products produce any scalar grades. Except for the bivector product this can be done by inspection. For that part we utilize the identity

This leaves for the energy in the volume

We are left with a completely real expression, and one without any explicit Geometric Algebra. This does not look like the Harmonic oscillator Hamiltonian that was expected. A gauge transformation to eliminate and an observation about when equals zero will give us that, but first lets get the mechanical jobs done, and reduce the products for the field momentum.

Momentum

Now move on to (16). For the factors other than only the vector-bivector products can contribute to the scalar product. We have two such products, one of the form

and the other

The momentum in this volume follows by computation of

All the products are paired in nice conjugates, taking real parts, and premultiplication with gives the desired result. Observe that two of these terms cancel, and another two have no real part. Those last are

Taking the real part of this pure imaginary is zero, leaving just

I am not sure why exactly, but I actually expected a term with , quadratic in the vector potential. Is there a mistake above?

Gauge transformation to simplify the Hamiltonian.

In (20) something that looked like the Harmonic oscillator was expected. On the surface this does not appear to be such a beast. Exploitation of gauge freedom is required to make the simplification that puts things into the Harmonic oscillator form.

If we are to change our four vector potential , then Maxwell’s equation takes the form

which is unchanged by the addition of the gradient to any original potential solution to the equation. In coordinates this is a transformation of the form

and we can use this to force any one of the potential coordinates to zero. For this problem, it appears that it is desirable to seek a such that . That is

Or,

With such a transformation, the and cross term in the Hamiltonian (20) vanishes, as does the term in the four vector square of the last term, leaving just

Additionally, wedging (5) with now does not loose any information so our potential Fourier series is reduced to just

The desired harmonic oscillator form would be had in (26) if it were not for the term. Does that vanish? Returning to Maxwell’s equation should answer that question, but first it has to be expressed in terms of the vector potential. While , the lack of an component means that this can be inverted as

The gradient can also be factored scalar and spatial vector components

So, with this gauge choice the bivector field is

From the left the gradient action on is

and from the right

Taking the difference we have

Which is just

For this vacuum case, premultiplication of Maxwell’s equation by gives

The spatial bivector and trivector grades are all zero. Equating the remaining scalar and vector components to zero separately yields a pair of equations in

If the divergence of the vector potential is constant we have just a wave equation. Let’s see what that divergence is with the assumed Fourier representation

Since , there are two ways for . For each there must be a requirement for either or . The constant solution to the first equation appears to represent a standing spatial wave with no time dependence. Is that of any interest?

The more interesting seeming case is where we have some non-static time varying state. In this case, if , the second of these Maxwell’s equations is just the vector potential wave equation, since the divergence is zero. That is

Solving this isn’t really what is of interest, since the objective was just to determine if the divergence could be assumed to be zero. This shows then, that if the transverse solution to Maxwell’s equation is picked, the Hamiltonian for this field, with this gauge choice, becomes

How does the gauge choice alter the Poynting vector? From (21), all the dependence in that integrated momentum density is lost

The solutions to Maxwell’s equation are seen to result in zero momentum for this infinite periodic field. My expectation was something of the form , so intuition is either failing me, or my math is failing me, or this contrived periodic field solution leads to trouble.

Conclusions and followup.

The objective was met, a reproduction of Bohm’s Harmonic oscillator result using a complex exponential Fourier series instead of separate sine and cosines.

The reason for Bohm’s choice to fix zero divergence as the gauge choice upfront is now clear. That automatically cuts complexity from the results. Figuring out how to work this problem with complex valued potentials and also using the Geometric Algebra formulation probably also made the work a bit more difficult since blundering through both simultaneously was required instead of just one at a time.

This was an interesting exercise though, since doing it this way I am able to understand all the intermediate steps. Bohm employed some subtler argumentation to eliminate the scalar potential upfront, and I have to admit I did not follow his logic, whereas blindly following where the math leads me all makes sense.

As a bit of followup, I’d like to consider the constant case in more detail, and any implications of the freedom to pick .

The general calculation of for the assumed Fourier solution should be possible too, but was not attempted. Doing that general calculation with a four dimensional Fourier series is likely tidier than working with scalar and spatial variables as done here.

Now that the math is out of the way (except possibly for the momentum which doesn’t seem right), some discussion of implications and applications is also in order. My preference is to let the math sink-in a bit first and mull over the momentum issues at leisure.

Motivation.

We now know how to formulate the energy momentum tensor for complex vector fields (ie. phasors) in the Geometric Algebra formalism. To recap, for the field , where and may be complex vectors we have for Maxwell’s equation

This is a doubly complex representation, with the four vector pseudoscalar acting as a non-commutatitive imaginary, as well as real and imaginary parts for the electric and magnetic field vectors. We take the real part (not the scalar part) of any bivector solution of Maxwell’s equation as the actual solution, but allow ourself the freedom to work with the complex phasor representation when convenient. In these phasor vectors, the imaginary , as in , is a commuting imaginary, commuting with all the multivector elements in the algebra.

The real valued, four vector, energy momentum tensor was found to be

To supply some context that gives meaning to this tensor the associated conservation relationship was found to be

and in particular for , this four vector divergence takes the form

relating the energy term and the Poynting spatial vector with the current density and electric field product that constitutes the energy portion of the Lorentz force density.

Let’s apply this to calculating the energy associated with the field that is periodic within a rectangular prism as done by Bohm in [1]. We do not necessarily need the Geometric Algebra formalism for this calculation, but this will be a fun way to attempt it.

Setup

Let’s assume a Fourier representation for the four vector potential for the field . That is

where summation is over all wave number triplets . The Fourier coefficients are allowed to be complex valued, as is the resulting four vector , and the associated bivector field .

Fourier inversion follows from

but only this orthogonality relationship and not the Fourier coefficients themselves

will be of interest here. Evaluating the curl for this potential yields

We can now form the energy density

With implied summation over all repeated integer indexes (even without matching uppers and lowers), this is

The grade selection used here doesn’t change the result since we already have a scalar, but will just make it convenient to filter out any higher order products that will cancel anyways. Integrating over the volume element and taking advantage of the orthogonality relationship (6), the exponentials are removed, leaving the energy contained in the volume

First reduction of the Hamiltonian.

Let’s take the products involved in sequence one at a time, and evaluate, later adding and taking real parts if required all of

The expectation is to obtain a Hamiltonian for the field that has the structure of harmonic oscillators, where the middle two products would have to be zero or sum to zero or have real parts that sum to zero. The first is expected to contain only products of , and the last only products of .

While initially guessing that (13) and (14) may cancel, this isn’t so obviously the case. The use of cyclic permutation of multivectors within the scalar grade selection operator plus a change of dummy summation indexes in one of the two shows that this sum is of the form . This sum is intrinsically real, so we can neglect one of the two doubling the other, but we will still be required to show that the real part of either is zero.

Lets reduce these one at a time starting with (12), and write temporarily

So the first of our Hamiltonian terms is

Note that summation over is still implied here, so we’d be better off with a spatial vector representation of the Fourier coefficients . With such a notation, this contribution to the Hamiltonian is

To reduce (13) and (13), this time writing , we can start with just the scalar selection

Thus the contribution to the Hamiltonian from (13) and (13) is

Most definitively not zero in general. Our final expansion (15) is the messiest. Again with for short, the grade selection of this term in coordinates is

Expanding this out yields

The contribution to the Hamiltonian from this, with , is then

A final reassembly of the Hamiltonian from the parts (17) and (18) and (21) is then

This is finally reduced to a completely real expression, and one without any explicit Geometric Algebra. All the four vector Fourier vector potentials written out explicitly in terms of the spacetime split , which is natural since an explicit time and space split was the starting point.

Gauge transformation to simplify the Hamiltonian.

While (22) has considerably simpler form than (11), what was expected, was something that looked like the Harmonic oscillator. On the surface this does not appear to be such a beast. Exploitation of gauge freedom is required to make the simplification that puts things into the Harmonic oscillator form.

If we are to change our four vector potential , then Maxwell’s equation takes the form

which is unchanged by the addition of the gradient to any original potential solution to the equation. In coordinates this is a transformation of the form

and we can use this to force any one of the potential coordinates to zero. For this problem, it appears that it is desirable to seek a such that . That is

Or,

With such a transformation, the and cross term in the Hamiltonian (22) vanishes, as does the term in the four vector square of the last term, leaving just

Additionally, wedging (5) with now does not loose any information so our potential Fourier series is reduced to just

The desired harmonic oscillator form would be had in (27) if it were not for the term. Does that vanish? Returning to Maxwell’s equation should answer that question, but first it has to be expressed in terms of the vector potential. While , the lack of an component means that this can be inverted as

The gradient can also be factored scalar and spatial vector components

So, with this gauge choice the bivector field is

From the left the gradient action on is

and from the right

Taking the difference we have

Which is just

For this vacuum case, premultiplication of Maxwell’s equation by gives

The spatial bivector and trivector grades are all zero. Equating the remaining scalar and vector components to zero separately yields a pair of equations in

If the divergence of the vector potential is constant we have just a wave equation. Let’s see what that divergence is with the assumed Fourier representation

Since , there are two ways for . For each there must be a requirement for either or . The constant solution to the first equation appears to represent a standing spatial wave with no time dependence. Is that of any interest?

The more interesting seeming case is where we have some non-static time varying state. In this case, if for all the second of these Maxwell’s equations is just the vector potential wave equation, since the divergence is zero. That is

Solving this isn’t really what is of interest, since the objective was just to determine if the divergence could be assumed to be zero. This shows then, that if the transverse solution to Maxwell’s equation is picked, the Hamiltonian for this field, with this gauge choice, becomes

Conclusions and followup.

The objective was met, a reproduction of Bohm’s Harmonic oscillator result using a complex exponential Fourier series instead of separate sine and cosines.

The reason for Bohm’s choice to fix zero divergence as the gauge choice upfront is now clear. That automatically cuts complexity from the results. Figuring out how to work this problem with complex valued potentials and also using the Geometric Algebra formulation probably also made the work a bit more difficult since blundering through both simultaneously was required instead of just one at a time.

This was an interesting exercise though, since doing it this way I am able to understand all the intermediate steps. Bohm employed some subtler argumentation to eliminate the scalar potential upfront, and I have to admit I did not follow his logic, whereas blindly following where the math leads me all makes sense.

As a bit of followup, I’d like to consider the constant case, and any implications of the freedom to pick . I’d also like to construct the Poynting vector , and see what the structure of that is with this Fourier representation.

A general calculation of for an assumed Fourier solution should be possible too, but working in spatial quantities for the general case is probably torture. A four dimensional Fourier series is likely a superior option for the general case.

Motivation

The problem of a solving for the relativistically correct trajectories of classically interacting proton and electron is one that I’ve wanted to try for a while. Conceptually this is just about the simplest interaction problem in electrodynamics (other than motion of a particle in a field), but it is not obvious to me how to even set up the right equations to solve. I should have the tools now to at least write down the equations to solve, and perhaps solve them too.

Familiarity with Geometric Algebra, and the STA form of the Maxwell and Lorentz force equation will be assumed. Writing for the Faraday bivector, these equations are respectively

The possibility of self interaction will also be ignored here. From what I have read this self interaction is more complex than regular two particle interaction.

With only Coulomb interaction.

With just Coulomb (non-relativistic) interaction setup of the equations of motion for the relative vector difference between the particles is straightforward. Let’s write this out as a reference. Whatever we come up with for the relativistic case should reduce to this at small velocities.

Fixing notation, lets write the proton and electron positions respectively by and , the proton charge as , and the electron charge . For the forces we have

FIXME: picture

Subtracting the two after mass division yields the reduced mass equation for the relative motion

This is now of the same form as the classical problem of two particle gravitational interaction, with the well known conic solutions.

Using the divergence equation instead.

While use of the Coulomb force above provides the equation of motion for the relative motion of the charges, how to generalize this to the relativistic case is not entirely clear. For the relativistic case we need to consider all of Maxwell’s equations, and not just the divergence equation. Let’s back up a step and setup the problem using the divergence equation instead of Coulomb’s law. This is a bit closer to the use of all of Maxwell’s equations.

To start off we need a discrete charge expression for the charge density, and can use the delta distribution to express this.

Picking a volume element that only encloses one of the respective charges gives us the Coulomb law for the field produced by those charges as above

Here and denote the electric fields due to the proton and electron respectively. Ignoring the possibility of self interaction the Lorentz forces on the particles are

In symbols, this is

If we were to substitute back into the volume integrals we’d have

It is tempting to take the differences of these two equations so that we can write this in terms of the relative acceleration . I did just this initially, and was surprised by a mass term of the form instead of reduced mass, which cannot be right. The key to avoiding this mistake is the proper considerations of the integration volumes. Since the volumes are different and can in fact be entirely disjoint, subtracting these is not possible. For this reason we have to be especially careful if a differential form of the divergence integrals (9) were to be used, as in

The domain of applicability of these equations is no longer explicit, since each has to omit a neighborhood around the other charge. When using a delta distribution to express the point charge density it is probably best to stick with an explicit integral form.

Comparing how far we can get starting with the Gauss’s law instead of the Coulomb force, and looking forward to the relativistic case, it seems likely that solving the field equations due to the respective current densities will be the first required step. Only then can we substitute that field solution back into the Lorentz force equation to complete the search for the particle trajectories.

Relativistic interaction.

First order of business is an expression for a point charge current density four vector. Following Jackson [1], but switching to vector notation from coordinates, we can apparently employ an arbitrary parametrization for the four-vector particle trajectory , as measured in the observer frame, and write

Here is the four vector event specifying the spacetime position of the current, also as measured in the observer frame. Reparameterizating in terms of time should get us back something more familiar looking

Note that the scaling property of the delta function implies . With the split of the four-volume delta function , where , we have an explanation for why Jackson had a factor of in his representation. I initially thought this factor of was due to CGS vs SI units! One more Jackson equation decoded. We are left with the following spacetime split for a point charge current density four vector

Comparing to the continuous case where we have , it appears that this works out right. One